Nowadays in silico analysis tools in the bio-medical field are moving from the research context to patient-specific treatment and prevention. Hemo-dynamics is receiving great attention, and accurate Computational Fluid Dynamics (CFD) modelling can be adopted to produce a digital medical twin capable of reliably predicting the pathology evolution, and the effect of surgical corrections. In this context, the availability of in silico digital twins based on computer-assisted engineering (CAE) simulations is one of the key enablers, and the parametric shape of vessels and reduced-order models (ROM) are promising solutions.
The Model Reduction carried out by ROM techniques allows cost-efficient evaluation of systems with a large degree-of-freedom (DOF) by reducing the number of variables involved in the problem. This simplification preserves the essential characteristics of the system and makes real-time control over the parameters of the problem feasible. This ROM peculiarity can be synergistically combined with mesh-morphing techniques that can provide a vast number of geometrical variations, by exploiting the morphing tool offered by Radial Basis Functions (RBF).
The solution of a radial basis functions (RBF) problem consists of calculating the coefficients of a linear system of order equal to the number of source points, by means of which the displacement of an arbitrary mesh node (target) can be expressed, and then imposed, as the summation of the radial contribution of each controlled node (source). In this way, mesh smoothing can be applied rapidly by maintaining the mesh topology, namely with the same type and number of elements.
The application selected to describe the proposed workflow is an ascending aorta aneurysm, studied using the new ROM Builder available in ANSYS 19.1. First, a patient-specific geometry was reconstructed, then a CFD model was created with a bulge shape that was parameterised using an RBF mesh morphing technique. Finally, a ROM was suitably built up by performing CFD simulations. We provided examples of fast evaluations that were achieved off-line by using the ROM results.
The 3D-model was obtained from the statistical analysis of the aortic morphological shapes of 45 aneurysmatic patients (Fig.1a - Statistical aneurysmatic aorta) whose Computed tomography (CT) datasets were retrospectively segmented and elaborated using VMTK software and Python custom scripts. The standard branch stent graft (SSG) model was parameterized to create a CFD model that could cope with many ascending aneurysmatic aortic configurations through shape deformations. Three of the parameters evaluated in the previous statistical analysis work were considered to follow the bulge variations: the maximum diameter of the aneurysm (Dmax), the tortuosity value (T) of the centreline of the ascending portion of aorta, and the bulge extension (Bext). These parameters are graphically shown in Fig.1b (Statistical aneurysmatic aorta) and were assumed to vary at specific intervals that were extracted from the results of a statistical analysis.
The mesh morphing action was accomplished using the commercial morpher RBF Morph™. A set of source points on the ascending thoracic aortic region was selected to generate five shape-variations representative of a statistical geometry. Three sets of RBF points were generated, as shown in Figure 2a (RBF set-up. Source points and their displacement), by maintaining fixed the extremal sets (green and red in the picture) in order to circumscribe the morphing action. The central set was translated into the plane over which the points were defined (Figure 2b, 2c and 2d - RBF set-up. Source points and their displacement), and was scaled over the same plane along the principal directions (Figure 2e and 2f - RBF set-up. Source points and their displacement). The time required to morph the 1.5 million element tetrahedral CFD mesh for each DP was 43s using an i7 laptop with 16 GB of RAM.
The ROM extraction process required the evaluation of a series of 40 high-fidelity snapshots selected via the Kriging method to capture all the important characteristics of the system, to seed the design points in the parameter space. The first five orthogonal modes were extracted with the Proper Order Decomposition (POD) to obtain a model that was reduced to 5 DOF.
To assess the quality of the proposed methodology, the results obtained using ROM were compared to those achieved using plain CFD evaluations. Figure 3 (ROM results (top) compared to CFD (bottom)) shows the pressure contours and streamlines obtained through ROM for the three degrees of accretion of the aneurism (in the first row), while the ones calculated using a full CFD computation are shown in the last row. The maximum discrepancy between the two methodologies is in the order of 2.5%. Although this error could be further decreased by enriching the number of snapshots and employing a higher number of orthogonal modes, it was judged to be acceptable because each new ROM evaluation could be visualised in almost real-time whereas the CFD calculations required about half an hour.
Since high-fidelity CFD simulations, which have already proven to be useful in predicting the growth and evolution of cardiovascular disease pathologies, require a lot of computational effort, a methodology to overcome this problem was proposed. ROM has proven to be very efficient in reducing the computing costs for complex multi degree-of-freedom models (like CFD models). On the other hand, RBF mesh morphing has proven to be highly efficient in generating different shapes for numerical models, thus reducing the time required to generate new ones. The proposed methodology envisages the synergetic use of ROM and RBF mesh morphing to generate a new shape for the CFD model including its numerical results from a discrete set of shape configurations.
In summary, the proposed methodology was developed in the ANSYS® Workbench™ 19.1 environment by exploiting the ROM functionalities provided by this release of the software together with the RBF Morph ™ Fluent® add-on. The complete workflow was tested in a CFD study of an ascending thoracic aortic aneurysm. The results obtained with the proposed approach were then compared to regular CFD evaluations, which required the full CFD solution of the new shapes of the aneurism. A good agreement in terms of the monitored variables was finally obtained. The successful application of the proposed methodology led to the conclusion that it can be successfully exploited to assist in the generation and fruition of digital twins in medical applications.